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Recent advances in stochastic PDEs, Hopf algebras of typed trees and integralequations have inspired the study of algebraic structures with replicatingoperations. To understand their algebraic and combinatorial nature, we firstuse rooted forests with multiple decoration sets to construct free Hopfalgebras with multiple Hochschild 1-cocycle conditions. Applying the universalproperty of the underlying operated algebras and the method ofGr\"obner-Shirshov bases, we then construct free objects in the category ofmatching Rota-Baxter algebras which is a generalization of Rota-Baxter algebrasto allow multiple Rota-Baxter operators. Finally the free matching Rota-Baxteralgebras are equipped with a cocycle Hopf algebra structure.

Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gr\"obner-Shirshov bases